Lagrangian density

The most fundamental quantity describing any physical system is its Lagrangian which contains information about the dynamics and symmetries of the system. The Lagrangian (density) for QCD can be given as

LQCD=^X

f

ψ¯_f(i /D−m_f)ψ_f−1

2Tr [F_µνF^µν], (3.1) where f denotes the quark flavor and the Feynman slash notation has been utilized — i.e.

D/ := γ^µD_µ. Here, γ^µ are the Dirac gamma matrices (see Appendix A for details). One should also notice that the Einstein summation convention has been used with repeated indices excludingf.

The first term represents quarks whereψ_f corresponds to quark field of flavorf, and it is a Dirac four-spinor. For QCD, whose (color-)gauge symmetry group is SU(3), everyψ_f has three components, in color space, corresponding to each color charge. The first term of Eq. (3.1) can be further split into two subterms: the kinetic and mass one, respectively. Here,m_f is the mass of the quark flavorf. In the kinetic term, the (gauge-)covariant derivative reads

D_µ:=∂_µ+igt_aA^a_µ, (3.2)

wheregis the strong coupling constant, theta’s are the generators of the Lie algebrasu(3) of the SU(3) group, and the A^a_µ’s are the color gauge fields of the gluons. Here, the index a is running from 1 to 8 representing the eight different gluons. In this case, the generators can be given using the Gell-Mann matricesλ_a(see Appendix A) so that

t_a=λ_a

2 . (3.3)

∗To be more specific, a gluon carries a superpositional combination of one color and one anticolor charges.

3.1 Basic properties 33

It can be shown that the Lagrangian is invariant under a local SU(3) gauge transformation:

ψ_f(x)7→U(x)ψ_f(x), (3.4)

A^a_µ(x)7→U(x)A^a_µ(x)U^†(x)− 1

igU(x)∂_µU^†(x), (3.5) where the unitary transformation function is

U(x) = exp [−itaθ^a(x)] (3.6) so that θ^a(x) is a function that depends on x. On the one hand, one can use the SU(3) symmetry as a starting point and to construct the QCD Lagrangian demanding that it contains all possible gauge-invariant terms up to mass dimension four. Following this procedure, one is able to create Eq. (3.1) with two additional modifications. Firstly, the Lagrangian contains another gluonic term — namely one proportional to^µναβTr (F_µνF_αβ), where^µναβ is the Levi-Civita symbol. Secondly, the quark-mass term should also contain a chiral phase parameterθ⁰ so thatm_f7→m_fexp(iγ₅θ⁰). On the other hand, experimental evidence [208, 209] suggests that these additional terms are very small or even nonexisting.

The second term of the QCD Lagrangian is the gluonic part, where the field strength tensor Fµνcan be given as

F_µν:=t_aF_µν^a = i

g[D_µ, D_ν]. (3.7)

Its components can be written as

F_µν^a =∂µA^a_ν−∂νA^a_µ+gsfabcA^b_µA^c_ν, (3.8) wherefabcis a totally antisymmetric structure constant (see Appendix A).

The feature that set QCD — or any non-Abelian field theory in general — apart from the simple QED framework with Abelian U(1) symmetry is the existence of the last term in Eq. (3.8).

Due to this contribution, the gauge bosons of non-Abelian theories — e.g. the gluons in QCD

— self interact. These interactions among three (3g) and four (4g) gluons can be presented as L3g=−g

2f^a_bc(∂^µA^ν_a−∂^νA^µ_a)A^b_µA^c_ν, (3.9) L4g=−g²

4fabcf^adeA^b_µA^c_νA^µ_dA^ν_e. (3.10) 3.1.2 Asymptotic freedom and the scale parameter

To examine the running of the couplingg, the so-called beta function can be defined as β(α_s) :=Qdα_s

dQ, (3.11)

whereQis the reference scale andαs:=g²/(4π). To date, the beta function of QCD has been calculated up to the five-loop order in the minimal subtraction (MS) scheme [210, 211]. The one-loop order result is given as

β(α_s) =2Nf−11Nc

6π α²_s+O(α³_s), (3.12)

whereN_f andN_care the number of quark flavors and colors, respectively [212, 213]. So far, six different flavors and three color charges have been detected, and notable signs of extra quark flavors (or colors) have not been found (see e.g. [80, 214–216]). If N_f/N_c < 11/2, the one-loop beta function is negative indicating that the coupling decreases with increasing energy — or with decreasing distance — asymptotically. This ultraviolet behavior of QCD is known as asymptotic freedom and it was discovered in 1973 [212, 213]. The beta function also suggests that high-density QCD phenomena can only be probed using perturbation methods because the coupling constantgis then small. QED, on the other hand, behaves differently — its one-loop beta function is strictly positive

β(α) =2α²

3π +O(α³), (3.13)

whereαis the fine-structure constant [217].

One can also reformulate the beta function using a separation of variables:

Z dQ

Q =^Z dαs

β(α_s). (3.14)

When solving the differential equation, then the dimensionful constant of integration can be expressed as

ΛQCD=Qexp− Z dα_s

β(αs)

. (3.15)

In the case of the one-loop-level result [Eq. (3.12)], this simply implies that α⁻¹_s (Q) =11N_c−2N_f

6π ln Q

ΛQCD

!

. (3.16)

Based on this formulation, it is easy to see that ΛQCD specifies the scale of the theory, and it is, therefore, known as the QCD scale parameter. If one wants to evaluate the value of ΛQCD, one can use the world average of the strong coupling constant

αs(mZ) = 0.1179±0.0010, (3.17) where the reference energy is equal to the mass of the Z boson,m_Z = 91.1876±0.0021 GeV [80].

This rough, one-loop-level estimation suggests that ΛQCD ≈200 MeV.

3.1.3 Phase diagram

One of the most interesting open problems in the study of QCD is to figure out the structure of the phase diagram. This diagram is usually illustrated as a function of the temperature T and baryon chemical potential µ_B (see Fig. 3.1). Even though the starting point — the Lagrangian of QCD — is well-known, any suitable first-principle method to probe the diagram at arbitrary temperatures and chemical potentials has not been developed yet. However, some useful approaches to tackle this problem already exist but their scope is very limited.

3.1 Basic properties 35

0 1 2 3 4

0 50 100 150 200

Baryon chemical potential(GeV)

Temperature(MeV)

A B

C D

Quark–gluon plasma

Hadrons

Cool quark matter

Figure 3.1: Schematic phase diagram of quantum chromodynamics. The three major phases

— the hadronic (blue), quark-gluon-plasma (yellow), and quark-matter (red) ones — are only shown. The solid lines correspond to the first-order phase transitions whereas the dashed lines represent crossovers. Likewise, the dots illustrate the critical points (A–C) and the triple point (D).

One of such method is perturbation theory. With this tool, it is possible to investigate the behavior of matter when the coupling of QCD is weak. Unfortunately, this implies that pertur-bative methods are only applicable when the temperature or density is large. Another widely used method is lattice field theory — anab initioapproach to study QCD nonperturbatively.

Due to the notorious sign problem [218], this method is only appropriate when the baryon chem-ical potentialµB is close to zero. Nevertheless, this region is one of the best-known part of the phase diagram at the moment (see details below). It is also possible to investigate the problem from a more phenomenological perspective using various model calculations (e.g. [219–224]).

Although this is an effective, qualitative way to scan the phase diagram, there is no guarantee that a model in hand captures the essential features of the underlying theory. So, the predictions of these models should always be treated with caution.

A schematic phase diagram of Fig. 3.1 illustrates the three main phases — the hadronic, quark-gluon-plasma, and quark-matter ones. Here, we will focus on these most notable phases because the phase diagram is still largely unknown. Nevertheless, it has been suggested that other possible phases — such the quarkyonic-matter one — may also exist, but we will not consider them in this thesis (see e.g. the review of [225] for further information). We will, however, present some interesting but speculative features of the diagram to stimulate discussion.

As hinted above, the behavior of the phase diagram is well understood at small chemical potentials. In this region, it has been observed that hadronic matter transforms into quark-gluon plasma via a smooth crossover (see Fig. 3.1) — a continuous change from one phase to another without distinguishable transition point — by increasing the temperature as lattice studies [226, 227] suggested. Even though the exact location of the crossover point cannot be pinpointed,

lattice studies [228–231] have been able to estimate the pseudo-critical temperature T_c based on the behaviors of several observables. These analyses propose that the value ofTc is about 155 MeV. In addition to these theoretical studies, collider experiments at CERN (European Organization for Nuclear Research) and BNL (Brookhaven National Laboratory) have managed to create quark-gluon plasma — plasma-like state consisting of deconfined quarks and gluons (see the detailed review of [232]).

Another well-studied part of the phase diagram is highlighted with purple color within the hadronic phase in Fig. 3.1. This corresponds to the first-order transition between the gaseous (lowµB) and liquid-like (highµB) hadronic phases. The corresponding critical point (marked as A) is located atn_c/n_s≈0.3–0.4 andT_c≈16–18 MeV. (See the review of [233].)

The third phase, shown in Fig. 3.1 as red, is the (cool) quark-matter one. Many model calculations have proposed that this part of the phase diagram consist of a collection of various color-superconducting phases (see e.g. the review of [234]; see also Fig. 2 of [220] or Figs. 1 and 2 of [235]). This kind of superconducting behavior is similar to the one with electrons described by the Bardeen–Cooper–Schrieffer (BCS) theory [236]. The more complex nature of the strong interaction, however, gives rise to a richer class of physical phenomena, i.e. the number of possible color-superconducting phases is huge. This does not necessarily have to be the case even though model calculations usually favor super-conducting phases. Although the (cool) quark-matter and quark-gluon-plasma phases are explicitly separated by a first-order transition (red solid line) in Fig. 3.1, it is certainly possible that these two phases form a united quark-matter phase together.

Beyond the above points, the phase structure given in Fig. 3.1 is even more speculative. For example, it is not known if the transition between the hadronic and quark-matter phases (blue solid line) is really a first-order one with a critical point B as the figure indicates (see e.g. [224]).

Nonetheless, multiple phenomenological studies, such as [219, 220, 222], support this claim.

Secondly, the existence of the third critical point (C) has also been hypothesized (e.g. [221, 223]) and some authors have constructed models with even more complex phase structure [237].

In a more typical scenario, the first-order-transition line meets theµ_B-axis without forming this so-called quark-hadron-continuity structure.

3.2 Nuclear physics at low densities

As pointed out in the previous chapter (see Fig. 3.2, as well), the nuclear-physics side of the EoS is well-known up to the crust-core interface — around 0.1 to 0.5 times the saturation density. For example, the works of [98, 99] are often used to describe this part of the EoS. The main degrees of freedom of nuclear matter are not quarks and gluons even though they are its fundamental building blocks. On the contrary, the effective degrees of freedom are confined configurations — such as the nuclei and pions. Hence, it is practical to use an effective-field-theory approach to study nuclear matter — especially near saturation. The most common approach is to utilize the

3.2 Nuclear physics at low densities 37

Figure 3.2: Known behavior of the equation of state of QCD at zero temperature. The relatively well-known outer- and inner-crust equations of state [98, 99] are illustrated using orange and blue colors, respectively. The low-density chiral-effective-field-theory results of [32] are highlighted with red color (Nucl.) while the high-density perturbative-QCD (pQCD) calculations [11, 238]

are given in black.

so-called chiral effective field theory (cEFT) which makes use of the almost chiral nature of the QCD Lagrangian. This can be seen from the chiral formulation of Eq. (3.1) with two lightest quark flavors

LQCD= ¯q_Li /Dq_L+ ¯q_Ri /Dq_R−q¯_Lmq˜ _R−q¯_Rmq˜ _L−1

2Tr [G_µνG^µν], (3.18) where ˜m is the mass matrix defined so that ˜m = diag(mu, m_d). Using the quark field q = (ψ_uψ_d)^T, the left- and right-handed component fields are given asq_L = (1−γ₅)q/2 andq_R= (1 +γ5)q/2, respectively. In the massless limit, the Lagrangian is invariant under the chiral transformation:

qi7→exp(−iθijσ^j/2)qi, (3.19) whereσ^jare the Pauli matrices (see Appendix A) andθ_ij are angles whilei=L, R. Compared to the nucleonic masses of O(GeV), the light-quark masses [O(MeV)] are insignificant, and therefore, the light-QCD Lagrangian is approximately chirally symmetric. [239]

In the cEFT framework, the calculations for pure neutron matter can be carried out using, e.g., perturbative methods, and currently, some contributions are even studied up to N⁴LO^∗ (see review [240] and reference therein). Here, the small perturbation parameters areP/Λcand m_π/Λc, whereP is the typical momentum scale andm_π ≈135 MeV is the pion mass, while Λc≈500 MeV is the breaking scale of the effective theory [35, 240]. The modern cEFT studies of neutron-star matter — such as [5, 35, 138] — often employ two- and three-nucleon interactions up to 2n_sat most because the uncertainties quickly increase with density. However, some more

∗Here, N⁴LO refers to the next-to-next-to-next-to-next-to-leading order.

ambitions calculations with distinctly higher densities (see e.g. [241]) have been performed as well. In this thesis, the conservative cEFT results of [32] for neutron-star matter up to 1.1ns

are used (cf. Fig. 3.2).

3.3 Perturbative QCD at high densities

In the weak coupling regime, where the temperature or the density are extremely large, pertur-bative techniques are available. Unfortunately, many interesting regions of the phase diagram — such as the cores of neutron stars — cannot be directly probed using these methods. In addition, some phenomena — such as the instantons — are not obtainable without a proper nonpertur-bative description. For an in-depth review of the current state of the EoS of perturnonpertur-bative QCD, refer to [242].

As has been pointed out, an old neutron star can be treated as an object at zero temperature.

In this limit with massless quarks, the full three-loop (α²_s) results of the EoS have been known for a long time [243] (see [244] as well), but recently, the first part of the four-loop contribution — namely theα³_sln²αsorder — has, besides, been solved [12]. It is anticipated that the four-loop result could be available — i.e. the remainingα³_slnα_sandα³_s terms will be computed — in the near future. On the other hand, a three-loop-level calculation for realistic strange-quark matter

— i.e. including the contribution of the strange-quark mass — has been carried out in [11]. A simple approximative formulation of these results ([238]; see the following subsection as well) has been used to model the high-density behavior of quark matter in the calculations described in this thesis (see Fig. 3.2).

Besides the zero-temperature limit, other perturbative regions have also been considered in the literature. Recently, calculations for dense matter with small, nonvanishing temperature have been carried out up toO(g⁵) [245]. With vanishing chemical potential and large temperature, the last fully perturbative (α³_slnα_s) order has already been derived [246] and the corresponding small-chemical-potential extension is also known [244].

3.3.1 Pocket formula

The three-loop zero-temperature pQCD results of [11] with nonzero strange-quark mass can be written in a more compact form as given in [238]. The pressure of beta-stable and electrically neutral QCD matter can be represented using only the baryon chemical potentialµ_B and the dimensionless scale parameterX:= 3¯Λ/µB:

ppQCD(µB, X) =pF D(µB)c1− a(X) (µ_B/GeV)−b(X)

, (3.20)

wherea(X) =d₁X^−ν1andb(X) =d₂X^−ν2. Here, ¯Λ is the renormalization scale — the arbitrary energy scale where the desirable renormalization conditions are assigned to remove unfavourable ultraviolet divergences — in the modified minimal subtraction (MS) scheme. Likewise, the

3.4 Holographic duality 39

function

p_{F D}(µ_B) = 3 4π²

µ_B 3

4

(3.21) is the Fermi–Dirac pressure which corresponds to the massless, noninteracting limit of a system of three quark flavors. The fitting parametersc₁,d₁,d₂,ν₁andν₂are selected so that the model faithfully represents the pressure, quark number density and squared speed of sound (see the original study for details). Whenµ_B>2 GeV^∗,p(µ_B)>0, and 1≤X≤4; the authors of [238]

argue that the optimal values for these parameters are

c₁= 0.9008, d₁= 0.5034, d₂= 1.452, ν₁= 0.3553, ν₂= 0.9101. (3.22)

3.4 Holographic duality

String theory is the leading candidate for a theory of quantum gravity. Although it has not been able to produce any observationally verifiable predictions, one of its side products — the gauge/gravity duality — has. Its main realization is the conjectured AdS/CFT correspondence

— proposed by J. Maldacena in 1997 [37] — which connects two very different kind of physical systems — (classical) gravity and quantum (field) theory. Here, AdS and CFT stand for anti–de Sitter space and conformal field theory, respectively, and these terms will be defined more closely in the following subsection.

In this thesis, the AdS/CFT approach has been exploited because it is a first-principle method to study various strongly coupled systems, such as QCD. As stated previously, there are noab initiomethods to probe the QCD directly at zero temperature. However, this kind of indirect way can possibly reveal important information about the generic behavior of strongly coupled systems. That is why it is significant to study this approach more closely.

3.4.1 AdS/CFT correspondence

Maybe the most famous and highly used version of the AdS/CFT correspondence is the setup where the gravity side is described by a ten-dimensional type IIB string theory on AdS5×S⁵ space whereas the corresponding dual field theory is the N = 4 supersymmetric Yang–Mills theory living in four-dimensional Minkowski space (see [37]). Here,S⁵ is the 5-sphere. This approach has been used in this thesis, but some other implementations of the correspondence have also been considered in the literature [37, 247].

The gravity side is mainly described by (d+ 1)-dimensional anti-de Sitter space (AdSd+1). It is the vacuum solution of the Einstein field equations with negative cosmological constant ΛC:

G_µν=−ΛCg_µν. (3.23)

∗Notice that the original research article [238] contains a typographical error. To fix it, the inequality symbol has been reversed.

A common way to represent the line element of AdS space is to use Poincaré coordinates:

ds²=L² z²

ηµνdx^µdx^ν−dz², (3.24) wherezis the extra dimension andLis the curvature radius while indicesµ, ν range from 0 to d−1. Variablesz andLare defined so thatz >0 and

L²=−d(d−1) 2ΛC

. (3.25)

Nevertheless, this parametrization does not cover the whole space, and hence, another suitable coordinate system has to be exploited at times, such as the so-called global coordinates. [248–

250]

The other main component of the correspondence is a CFT. It is a quantum field theory that is invariant under conformal transformations that leave all angles intact but allow, e.g., rotations and translations to happen. Often, theN = 4 supersymmetric Yang–Mills (SYM) theory is used to describe the quantum side of the duality because it resembles QCD and it is a (relatively) well understood theory. Although the particle content is, e.g, rather similar, these two theories differ at the level of details; unlike QCD, the SYM theory is a CFT and supersymmetric, for instance. Put it more precisely, the first feature basically means that the SYM theory cannot have massive particles because it would break the needed scale invariance. The second condition suggests that there are some kinds of (yet experimentally unobserved) symmetries that connect the degrees of freedom of bosons and fermions. Actually, the N = 4 condition underlines the fact that the given SYM theory is as supersymmetric as theoretically possible. [249–251]

So to speak, the CFT lives on the surface of AdS space meaning that the dimension of the AdS space is one larger than that of the corresponding CFT. This property follows the holographic principle which suggests that the surface of a system contains encoded information about its content [252, 253]. One of the most famous examples of the holographic principle is the fact that the entropy of a black hole is proportional to the surface areaABH— and not the volume

— of the event horizon: [254–256]

S=A_BH

4G . (3.26)

This statement also implies that a black hole should radiate as S. W. Hawking famously sug-gested [255, 256].

In this thesis, the AdS/CFT correspondence has been used because it has been conjectured that the dual of a strongly coupled SYM theory is a weakly coupled system described by classical gravity (andvice versa). Often, calculations are much easier to do in the weak limit, and these results can then be converted to describe the strongly coupled system using the duality together with a suitable dictionary. In this matter, one can get information about a QCD-like EoS by studying a simpler gravity-based system. [248, 249]

It is good to be aware that there exists several versions of the AdS/CFT conjecture. The weakest one states that the correspondence is only valid if the ’t Hooft coupling,

λ_YM:=g_YM² N_c=g_sN_c, (3.27)

3.4 Holographic duality 41

is large. Here, g_YM and g_s are the Yang-Mills and string couplings, respectively. A stronger version, in contrast, allows finite ’t Hooft coupling but the string coupling has to vanish when N_c → ∞. Moreover, the strongest formulation of the conjecture guarantees that the duality stands regardless of the values ofgs andNcif the dual system is solvable. [249]

It should be noted that the above introduction of the AdS/CFT correspondence is only a brief

It should be noted that the above introduction of the AdS/CFT correspondence is only a brief

In document From QCD to Neutron Stars and Back : Probing the Fundamental Properties of Dense Matter (sivua 40-0)